Question: Solve for $x$ : $ 2|x + 6| + 9 = 4|x + 6| + 10 $
Solution: Subtract $ {2|x + 6|} $ from both sides: $ \begin{eqnarray} 2|x + 6| + 9 &=& 4|x + 6| + 10 \\ \\ {- 2|x + 6|} && {- 2|x + 6|} \\ \\ 9 &=& 2|x + 6| + 10 \end{eqnarray} $ Subtract $10$ from both sides: $ \begin{eqnarray} 9 &=& 2|x + 6| + 10 \\ \\ {- 10} && {- 10} \\ \\ -1 &=& 2|x + 6| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{-1} {{2}} = \dfrac{2|x + 6|} {{2}} $ Simplify: $ -\dfrac{1}{2} = |x + 6| $ The absolute value cannot be negative. Therefore, there is no solution.